Verlag der Österreichischen Akademie der Wissenschaften

Abstract:
Optimized Schwarz methods enhance convergence using optimized transmission conditions between subdomains. The optimization is usually performed for a model problem on an unbounded domain and two subdomains represented by half spaces. The influence of the domain decomposition geometry on the convergence and the optimized parameters is thus lost in the process, and it is not even theoretically clear if the results published for the unbounded domain still hold in concrete applications where the domains are bounded. We prove here rigorously for a two-subdomain decomposition that the asymptotic performance of optimized Schwarz methods derived from an unbounded domain analysis still holds in the case of a bounded domain, but the constants in the best choice of parameters and convergence rate estimates are influenced by the domain truncation. We obtain accurate estimates for this influence and show theoretically that the domain truncation has more remarkable influence for the slowly converging optimized Schwarz methods than for those converging fast. When the subdomain size is very small, our new optimized parameters lead to much faster algorithms than those obtained from an unbounded domain analysis. We illustrate our theoretical results with numerical experiments.